Poll mentioned by The New York Times

There will be important primaries or runoff elections on Tuesday in seven states, including New York, where establishment candidates in both parties face tests in colorful, close primary battles.

A former congressman on Staten Island will try to regain his seat, hoping Republican voters look past his criminal record as a tax evader. On the Democratic side, challenges to incumbents have revealed tensions between party leaders backing entrenched candidates and new candidates who represent an infusion of activism in the party.

Voters will also go to the polls in Utah, Maryland, Colorado, Oklahoma, Mississippi and South Carolina.

They will be winnowing down the field of candidates from both parties who will face off in the midterm elections on Nov. 6. Here are a few things to watch for.

New York

Much of the attention on Tuesday will be on a Republican race on Staten Island, where Michael Grimm, a former congressman who pleaded guilty to felony tax fraud in 2014, is running to regain his seat in the 11th District.

Mr. Grimm is facing Representative Dan Donovan, the Republican who succeeded him. A victory by Mr. Grimm would give Democrats an opening in solidly Republican territory, as they will aggressively use Mr. Grimm’s criminal record against him.

Democrats plan to compete hard this fall across New York State, where they hope to pick up Republican-held House seats in regions as varied as the suburbs of Long Island and the farmland along the border with Canada. But crucial races Tuesday will also focus attention on intraparty competition involving four incumbents in and around New York City. Though they are favored to win, the incumbents have faced aggressive challenges from insurgent candidates.

The most closely watched race will be in the 14th District, where the 10-term incumbent Joseph Crowley has been challenged by Alexandria Ocasio-Cortez, a progressive candidate who has called for dismantling the Immigration and Customs Enforcement agency.

Also facing tough primary challengers in the metropolitan area: Representatives Eliot L. Engel, Yvette D. Clarke and Carolyn Maloney, who is running against Suraj Patel, a hotel executive, in the 12th District.

Elsewhere in the state, Democrats in the 24th District, which stretches from south of Syracuse to the shores of Lake Ontario, will pick a candidate to take on Representative John Katko, a Republican. The party-endorsed candidate, Juanita Perez Williams, a former Navy officer, is facing off against Dana Balter.

Seven Democrats are competing in a primary to run against Representative John Faso, a Republican, in the 19th District, comprising the Hudson Valley and the Catskill Mountains.

In the First District, on Long Island, voters will pick between five Democrats vying for the chance to challenge Representative Lee Zeldin, a Republican.

The 21st District, which hugs the Canadian border, has five Democrats who want the chance to compete against Representative Elise Stefanik, a Republican, in November.

Maryland

The Democratic primary is the race to watch in Maryland on Tuesday.

A crowded slate of Democrats is competing for the chance to challenge Gov. Larry Hogan, a popular Republican. They include Ben Jealous, the former president of the N.A.A.C.P., and Rushern Baker, the executive of Prince George’s County. They’re widely seen as the front-runners.

Senator Benjamin L. Cardin, a two-term incumbent who was once the ranking Democrat on the Senate Foreign Relations Committee, also faces a primary challenge.

Chelsea Manning, the former Army private who was convicted of leaking classified documents to WikiLeaks in 2013, is running against him.

Utah

Mitt Romney is the man to watch in Utah.

The former Massachusetts governor and 2012 Republican candidate for president is hoping to restart his political career with a Senate run in one of the deepest red states in the country. He wants to replace Orrin G. Hatch, who is retiring at the end of this term.

It hasn’t been totally smooth sailing for Mr. Romney, though. He must compete in the Republican primary because he failed to win more than 50 percent of the vote at the Utah Republican convention in April.

That embarrassing result was driven in part by questions about his commitment to Utah, where Mr. Romney helped plan the 2002 Salt Lake City Olympics but has never held elected office.

Nevertheless, Mr. Romney is expected to defeat his opponent, a state senator named Mike Kennedy.

ColoradoThe biggest race in the state is the contest to succeed Gov. John Hickenlooper, a moderate Democrat who is being term-limited out of office. Four candidates from each party are angling to replace him.

The race raises question about whether social and economic change — everything from rising home prices to legalized marijuana — may be turning Colorado, once reliably purple, into a more deeply blue state. The state’s 1.2 million independent voters can participate in party primaries for the first time this year.

The Democratic candidates include Representative Jared Polis; Cary Kennedy, a former state treasurer; and Mike Johnson, a former state senator. The Republican front-runner is Walker Stapleton, the state treasurer. He has campaigned as a supporter of President Trump, praising both his tax cuts and his opposition to sanctuary cities.

There is also an interesting Democratic race in the First District, in Denver, a seat Representative Diana DeGette has held for more than 20 years. Ms. DeGette, a member of the House Democratic leadership, is facing a left-wing primary challenge from Saira Rao, a former Wall Street lawyer and political newcomer who has denounced corporate influence on the Democratic Party.

Oklahoma

Oklahoma also has a closely watched governor’s race this year, but the state is so deeply red that the energy there is mostly focused on the Republican primary.

That has set up a contest between her lieutenant governor, Todd Lamb, and a string of challengers, including Mayor Mick Cornett of Oklahoma City, State Auditor and Inspector Gary Jones and Kevin Stitt, a banker.

Voters will also decide on State Question 788, a proposal that would legalize medical marijuana for users who obtain a license.

South CarolinaGov. Henry McMaster, who stepped into the governor’s mansion after Nikki Haley joined the Trump administration last year, was forced into a runoff when he failed to win more than 50 percent of the vote in the Republican primary last week. The governor has aligned himself closely with President Trump and got a boost from the president at a rally in Columbia on Monday evening.

Mr. McMaster, who won roughly 42 percent of the primary vote, will go up against John Warren, a businessman who styles himself as a Trump-like political outsider. Mr. Warren won around 28 percent of the primary vote.

MississippiMississippi will hold a Democratic runoff on Tuesday to determine who will challenge Roger Wicker, the state’s junior senator, in the midterm elections. But it is one of the most solidly Republican states in the country, making Mr. Wicker’s re-election appear all but guaranteed.